A differentiation index for nonlinear partial differential-algebraic equations is presented. Determination of the differentiation index with respect to a direction in the space of independent variables uncovers all equations that must be satisfied by {Cauchy} data on the hyperplane orthogonal to that direction. This index is thus a generalization of the differentiation index of differential-algebraic equations. The motivation for this work is consistent initialization of partial differential-algebraic systems. The analysis is demonstrated through several examples, including some taken from fluid dynamics and chemical engineering.